For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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How to parametrize a cyclic quadrilateral

In hopes of parametrizing some variables (either lengths, or angles, or both) of a cyclic quadrilateral, I was looking for a rule, or a set of rules defining a cyclic quadrilateral in terms of these ...
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1answer
19 views

Finding the base of trapezoid using diagonals and the angle between them. diagonals length 6 and 8, the angle is 90 degrees.

Find the large base of trapezoid, with diagonals length 6 and 8, if the angle between them is 90 degrees. Actually the original problem asks to find the length of the line parallel to the base, that ...
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4answers
25 views

Proof for primitive pythagorean triples

I was wondering if there was a proof that any integer pythagorean triple can be represented as a positive integer multiple of a primitive pythagorean triple. This seems quite related to the ...
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1answer
37 views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
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5answers
63 views

Determining the distance of a point from a line segment given a starting and ending point

I've found a lot of answers on how to find the distance from a point to a line, but not so much from a point to a line segment. I am given the $x$ and $y$ coordinates of the start point and end ...
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1answer
49 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
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1answer
29 views

How to find the area of overlap of two quarter circles

The question is to find the area of a shaded region. The shaded region is the overlap of two quarter circles both of which have two of their radii on the edge of a 15x15cm square. Both quarter circles ...
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0answers
19 views

Prove/Disprove Theorem about regular polygons

Given any regular polygon, and a point inside the polygon, prove that the sum of the shaded areas that formed of the point, vertices and altitudes, are equal to the some of the unshaded ones. ...
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1answer
112 views

A connectivity-preserving function from a connected set onto an interval

Let $C$ be a connected set in the plane and $I$ the unit interval interval. Call a function $f$ from $C$ onto $I$ Connectivity-preserving if the following is true for every subset $I'\subseteq I$: ...
3
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1answer
69 views

Circle is similar to a polygon with infinite number of sides

It is know from the time of Euclid, that a circle is similar to a polygon with infinite number of sides. But this ^^ is informal. Do you know any formalization where it appears that a circle is a ...
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3answers
71 views

Distance between a circle and a line

Find the distance between the circle $(x-3)^2+(y+2)^2=4$ and the line $x + 2y = 9$.
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1answer
25 views

What is the general equation of finding position x of images when they are aligned at center?

Suppose I have an image a1 and the anchor point is at center of image, when it is aligned at centre, the position is 0,0 When another image a2 is added and a1 and a2 together are aligned at center, ...
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0answers
51 views

An alternative derivation of radius of curvature (2D functions). How valid is it?

I was wondering how radius of curvature was derived, and this is what I came up with. It turned out to be longer than expected. Then I looked at how it compares with other (presumably more ...
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0answers
20 views

Values of collinear points

Points $A, B,$ and $C$ are collinear and $AB$:$AC$$=\frac{2}{5}$. Point $A$ is located at $(-3, 6)$, point $B$ is located at $(n, q)$, and point $C$ is located at $(-3, -4)$. What are the values of ...
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0answers
40 views

Pythagorean theorem question

In an isosceles triangle, the length of each leg is $13$ and the length of the base is $24$. What is the length of the altitude drawn to the base?
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1answer
28 views

Geometric Meaning behind the algorithm (slope of the line + ray casting)

I'm trying to dissect the classic algorithm for finding if a point is inside a (simple) polygon. Please see: http://erich.realtimerendering.com/ptinpoly/ and ...
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1answer
31 views

Proof concerning isosceles triangles

In the triangle $ABC$ it is $AC = BC$ and $\alpha = \beta$. The points $D$ and $E$ are on the line through $A$ and $B$. Show that the triangle $CDE$ is isosceles. Hey there! Is it sufficient ...
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1answer
73 views

Three random points in a cube

Select three points $A$, $B$, and $C$ randomly from a uniform distribution in a cube. What is the probability that $C$ is inside the sphere whose diameter is $AB$? This problem might be pretty messy ...
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1answer
33 views

Does $(x,y,z) = (2,1,1) +s(-1,-1,-1) + t(2,-2,-2)$ represent a line or plane?

Does the equation $$(x,y,z) = (2,1,1) +s(-1,-1,-1) + t(2,-2,-2)$$ represent a line or plane? I claimed it is a plane, as the two direction vectors are not multiples and thus for any values of $s$ and ...
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1answer
19 views

Relation between the lenghts of unequal chords and their distances from the centre

Is there any proportional relation between the lenghts of unequal chords and their respective distances from the centre of a circumference?
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2answers
56 views

Find intersection point of 3 circles

so first of all, I just want to point out that I am a beginner, so cut me some slack. As the title says I have 3 circles. I know the coordinates of each center and the radius of each circle. What I ...
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3answers
50 views

Prove a parallelogram inside parallelogram

I have drawn a figure, In parallelogram ABCD, AP is the bisector of angle A CQ is the bisector of angle C Can I prove APCQ is a parallelogram? or it isn't? I first joined AC and now if somehow I ...
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2answers
38 views

I need some help with Geometry. Is this a correct answer to this problem?

Good day, I have a question regarding geometry. I don't know whether my answer is correct because the answer in my book uses a totally different method for solving this particular problem. Here's ...
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0answers
33 views

Variation of the opaque forest problem (a.k.a farmyard problem)

I was wondering about the following variation of the opaque forest problem (see here and there for previous questions) : What is the least length set of segments that will intersect every straight ...
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3answers
43 views

Distance of centroid to incenter

Suppose there is a right triangle where all side-lengths are integers. The distance from the circumcenter to the centroid of the triangle is 6.5. Find the distance from the centroid to the incenter ...
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1answer
35 views

Simultaneous equation with summation and square - how to solve?

$\mathbf{p}$ is a vector with dimension: $x \times 1$ $\mathbf{d}$ is a vector with dimension: $1 \times y$ $\mathbf{V}$ is a matrix with dimension: $x \times y$ $y \geq x$ $\mathbf{d}$ and ...
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1answer
32 views

Dividing Up A Circular Search Area

BACKSTORY: I need to collect 500 plant samples for strontium analysis. The samples are randomly distributed across a circular area with a radius of 300 kilometers. I have to do this in 30 days, so I ...
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0answers
28 views

Specifications for concave polygons

What makes a polygon concave or convex? We all know that convex polygons have angles less than $180^{\circ}$ and concave polygons don't, but what really makes a polygon concave? In other words, what ...
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1answer
34 views

Length of hypotenuse of a right triangle when dimensions are not scaled equally

What I ask is if $1$ meter in $x$ direction is $2$ times bigger than $1$ meter in $y$ direction. What is the length of hypotenuse when for ex, $3$ in $x$ direction and $4$ in $y$ direction ? I ...
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0answers
17 views

is this sufficient to define a simplex?

I want to define a simplex based on the following properties A convex polytope All vertexes share an edge with all others For a given vertex $v_i$ the set of all facets that the vertex belongs to ...
2
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1answer
107 views

Which of the $43,380$ possible nets for a dodecahedron is the narrowest?

I want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width. My hunch is that the ...
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0answers
33 views

How to calculate the volume of an arbitrary pyramid without calculus?

I've been reading about the intuition behind calculating the volume of a pyramid by dividing the unit cube into 6 equal pyramids with lines from the center of the cube and it makes sense since all ...
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1answer
31 views

Find sides of a right triangle given hypotenuse c and area A (no numbers given)

I've solved couple of these, but I have no idea how to solve it without any numbers provided. I've tried using $A=\frac{ab}{2} \Rightarrow 2A=ab \Rightarrow 4A^2=a^2b^2$ and incorporating ...
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1answer
24 views

Regular Triangulations of Cube

I want to figure out which triangulations of the cube (i.e., partitions into tetrahedra using only the $8$ given vertices) are regular, but I'm not sure how to easily tell whether a given ...
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0answers
51 views

Creating Fundamental Groups and How are they Described?

Im a math learner so this question may seem obvious. Consider the fundamental group of a torus (we call this the object $O(1)$). Suppose we have another torus and can glue it to the other torus to ...
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2answers
60 views

Show that among all triangles with fixed $s$ and $a$, the area is maximised when $b=c$.

Given a triangle $ABC$, let $a= \bigl| BC \bigr|$, $b= \bigl| AC \bigr|$ and $c= \bigl| AB \bigr|$ and let $s=\frac{1}{2}(a+b+c)$ be the semiperimeter. (a) Show that among all triangles with fixed ...
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0answers
33 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
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1answer
39 views

${2\pi \over 3} = 2u + \sin {2u}$ (intersections of circles)

So, I was browsing the internet today, when I saw an interesting problem: Two circles, each with radii of one, are intersecting. If the area enclosed by the intersection of the two circles is equal to ...
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1answer
47 views

Question on Geometry, triangles and circles

Let $ABC$ be a right-angled triangle with $B = 90^\circ$ and let $BD$ be the altitude from $B$ on to $AC$. Draw $DE$ perpendicular to $AB$ and $DF$ perpendicular to $BC$. Let $P, Q, R$ and $S$ be ...
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2answers
25 views

Relation between areas of two regular polygons, one with twice as many sides [closed]

I am stuck with the following exercise given on Spivak's Calculus (chapter 8, exercise 11-b): Suppose $P$ is a regular polygon inscribed inside a circle. If $P'$ is the inscribed regular polygon with ...
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1answer
24 views

Pairwise tangent circles radical axes

Three pairwise tangent circles are drawn with the three common tangents to each of the pairs of circles. Prove that the common tangents must intersect at a point. Since the tangents to the ...
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1answer
44 views

Why is it called $SSS$ similarity?

Are two triangles with two sides in proportion automatically similar? If so, why is the postulate called $SSS$ similarity?
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2answers
25 views

Find the sum of the areas of all rectangles whose area is tripled when three units are added to the height and two units are added to the length

A rectangle has all sides of integer length. When three units are added to the height and two units to the length, the area of the rectangle is tripled. What is the sum of all the original areas of ...
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1answer
31 views

Calculating cross section of rectangle by angle

Given this: How do you calculate the length of the green line, given x degrees, and the fact that height / width = 2 / 5? The blue line indicates at 0 degrees. The length of the pink line equals ...
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1answer
29 views

Prove that $\angle I_aB_0I_c=90$

Given triangle $ABC$ and point $D$ on $AC$. Let $I_a, I_c$ be the centers of inscribed circles of $ABD$ and $BCD$ respectively. $B_0$ is the point, incircle of $ABC$ touches $AC$. Prove that ...
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1answer
44 views

Is the matrix filled with the areas of pairwise intersections of disks in a plane always positive semidefinite?

Consider disks $s_1, \cdots, s_n$ in the plane and let $a_{ij}$ be the area of $s_i\cap s_j$. Is it true that for any real numbers $x_1,\cdots, x_n$ we have $$ \sum_{i,j=1}^n x_ix_j a_{ij} \geq 0$$ ...
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0answers
21 views

What is the solid of revolution of an ellipsis around a general line through the origin?

If one rotates an ellipsis around its major axis, one gets a prolate ellipsoid. Around the minor axis, ones gets an oblate ellipsoid. What about a general line through the origin?
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1answer
58 views

Motivation for studying rational curves

Why do we study rational curves? A curve $f(x,y)=0$ is called a rational curve if there exists two rational functions $\chi(t)$ and $\psi(t)$ such that $f(\chi(t),\psi(t))=0$ for all $t$. Why is it ...
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2answers
31 views

What types of triangles are constructible?

What types of triangles are constructible? I know that equilateral triangles are easily constructed using compass and straightedge, but what about other types of triangles? Can any other ...
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4answers
24 views

Feynman lectures, Volume I, chapter 13-4

While reading Feynman lectures on Physics, volume I, Chapter 13-4, I found following assumption, which I don't understand: Then, since $r^2 = \rho^2 + a^2$, $\rho\,d\rho = r\,dr$. Therefore ... ...